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0Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 473-480. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220701
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70-01, 70H03, 70H33
On a Change of Variables in Lagrange's Equations
A. P. Markeev
This paper studies a material system with a finite number of degrees of freedom the motion of which is described by differential Lagrange's equations of the second kind. A twice continuously differentiable change of generalized coordinates and time is considered. It is well known that the equations of motion are covariant under such transformations. The conventional proof of this covariance property is usually based on the integral variational principle due to Hamilton and Ostrogradskii. This paper gives a proof of covariance that differs from the generally accepted one.
In addition, some methodical examples interesting in theory and applications are considered. In some of them (the equilibrium of a polytropic gas sphere between whose particles the forces of gravitational attraction act and the problem of the planar motion of a charged particle in the dipole force field) Lagrange's equations are not only covariant, but also possess the invariance property.
Keywords: analytical mechanics, Lagrange's equations, transformation methods in mechanics
1. Introduction
Consider a material system with n degrees grange's equations of the second kind
d_dL_ dL_ _ dt d^ dqi
Received May 31, 2022 Accepted July 12, 2022
This research was carried out within the framework of the state assignment (registration No. AAAA-A20-120011690138-6) at the Ishlinskii Institute for Problems in Mechanics, RAS, and at the Moscow Aviation Institute (National Research University).
Anatoly P. Markeev anat-markeev@mail.ru
Ishlinsky Institute for Problems in Mechanics RAS pr. Vernadskogo 101-1, Moscow, 119526 Russia, Moscow Aviation Institute (National research university) Volokolamskoe sh. 4, Moscow, 125993 Russia
of freedom whose motion is described by Lai = 1, 2,...,n. (1.1)
Here Qi (i = 1,2,..., n) are generalized coordinates, time t is an independent variable, q_i = -% (i = 1, 2, ..., n) are generalized velocities, and L is a Lagrangian function,
L = L(q^ ...,qn,ql,..., ^ t).
(1.2)
In the equations of motion (1.1), we transform from the "old", coordinates ql, ..., qn and the "old", time t to "new", coordinates ql, ...,qn and a "new", time variable t by the formulae
qi = qi(qi, ..., Qn,t), t = t^,...,^, t), i = 1, 2,...,n.
(1.3)
Assume that the functions qi and t on the right-hand sides of these formulae are twice continuously differentiable with respect to all variables qv ..., qn, t, and that their Jacobian J is nonzero:
J
P(Qi, •••, Qn, t) D(qv ..., qn, t)
9<h
dq1
d(in
dq1 dt dq1
9<h d(ii
dqn at
d(in d1n
dq„ at
at qi
dqn at
= 0.
(1.4)
If condition (1.4) is satisfied, equations (1.3) are uniquely solvable for the "new", variables, and ql, ..., qn, q are twice continuously differentiable with respect to the "old", variables ql, ..., qn t [1].
Let us call the quantities ^ (k = 1,2,..., n) generalized velocities corresponding to the new variables. Find the expression of the "old", generalized velocities iji in terms of the "new", generalized velocities Differentiating the second of Eqs. (1.3) with respect to t, we find the total derivative t with respect to the "new", time variable t:
dt ^ dt dqk dt dt ~ f-f dqk dt dt
(1.5)
k=l
Similarly, the first of Eqs. (1.3) gives the relation
fjf 2-^t f)n, rU '
k=l dqk dt
dt
i = 1, 2,
n.
(1.6)
From (1.5) and (1.6) we obtain the desired expressions for the generalized velocities q^
dt
dfg dt dt^ dt
V 4-
dik dt ^ at ^ dqk dt ^ at
i = 1, 2, ..., n.
(1.7)
Lagrange's equations possess the property of covariance under transformations of the form (1.3). This implies that under such transformations the form in which the equations of motion are written does not change. Transformations of the form (1.3) change only the La-grangian function: instead of the "old", Lagrangian function (1.2) in the transformed equations there will be a "new", function
L = ^q^ ..^q^
dql dqn
dq
dq
t
(1.8)
and the transformed equations of motion will have the same form as the initial equations (1.1):
d dL dL
0, j = 1, 2, ..., «. (1.9)
It is well known that in the transformed equations of motion (1.9) the function (1.8) can be given by the equation1
~ dt
L = L(ql, qn, qu ..., qn, t)-~, (1.10)
dt
where in the arguments of the function L the quantities qi and t are the right-hand sides of Eqs. (1.3), the quantities ^ are calculated from the formulae (1.7), and the factor on the right-hand side of Eq. (1.10) is given by relation (1.5).
Validation of the covariance property of Lagrange's equations and derivation of the formula (1.10) for the Lagrangian function of the transformed equations of motion are usually made using the Hamilton principle (see, e.g., [3-5]). The corresponding constructions are very simple and elegant. But direct verification of the validity of Eqs. (1.9) with the function L calculated by the formula (1.10) is also of interest. The corresponding validation in the case where time t is not transformed (i. e., in the transformation (1.3) t = t) can be found in the books [6, 7].
2. Auxiliary relations
The following identities hold:
¿,7 = 1,2,...,«. (2.1)
dqj dt dt dqj' dq^ dt dtdq^
These identities follow immediately from Eqs. (1.6) and (1.5) if in calculating the mixed second-order partial derivatives of the functions qi and t one takes into account their continuity and hence [1] the permutability of the order of differentiation with respect to the variables <f1, ..., qfn, t. Let us verify, for example, the validity of the first of identities (2.1). From Eqs. (1.6) we have
JLf^i = y^ 5)2% dqk 62qi = A d2^ dqk 62qi = d dqi
dq} dt dqk dq} dt dtdqj dqj dqk dt dqj dt, dtdq} '
We also note the following identities which will be useful in the sequel:
These identities are obtained by differentiation of the quantities q_i, given by (1.7), with respect to the variable
at
1 The spaced-out word "can" indicates, in particular, that the Lagrangian function is restored from the
equations of motion ambiguously. For example, the equations of motion will not change if one adds to the Lagrangian function the total time derivative of an arbitrary twice continuously differentiable function of generalized coordinates and time [2-4].
3. Covariance of Lagrange's equations
To validate the covariance property of Lagrange's equations, we write an expression for the left-hand sides of Eqs. (1.9). As a preliminary, we note that (1.10), (1.5) and (2.2) lead to the equations
dL
^ dL d^ dt dt ^ dL ( dqi . dt \ dt
dL I ^ dL dqi dL d^ dL dt \ dt d dt . dq - * ftn - ftn - ftà - ftn - ftt Fin. I Fin. rti1 ^ ' ' '
dqi dq- dq_-
dt dq- dq
dt
(3.1)
(3.2)
and the last term in (3.2), according to the second group of identities (2.1), can be replaced
by L-
©<? ;
dt
Hence,
d dL
dt
dL
dqi
dt
dL d i dqi
dt
dL d / dL \ / dq- . dt \ dL d / dq- . dt \
i=l
En ( dL dqi dL d^ dL dt \ dt d j dt
i=l
^3
I dL dqi dL d^ dL dt \ dt d dt ^ ydqi dq3 d^ dqj dt dqj j dt dq,• dt
^3
*3
En d ( dL\ dt ( dqi . dt \ ^ dL ( dqi dt dt dqi dt \
jt {^J di\W3~qiW3) \ dtdtW3~W3di) +
i=l
d ( dqi . dt
%~qw3
dt
dq_i dt dq_i dt dt dqj dqj dt
j = 1, 2,...,n. (3.3)
But the expression in the square brackets is identically zero. Indeed, performing necessary differentiations and using identities (2.1), we obtain the following chain of equalities:
d ( dqj dt \ dq-
dt_
+
dqi dt dq_i dt d I dqi
dq dq3 dq3 dq dt \ dqj
+
q dt dqi dt
dq dq3 dq3 dq
JL
)
d_
%
% dt
dt
1i
dqi dt
dt dq
d
3 dt
dq3 \dt
d_dt
■+
d
dt
9idt
JL
dq}
3
^ dt
lq3 dt %
dt
d f dq,
dq3 V dq
=0.
Therefore, relations (3.3) can be rewritten as
d dL
dL
E
i=l
d, dL dL\ dt I dqi
dt dqi dqi J dt I dq3
dt_
ldq,
j = 1, 2,
n.
(3.4)
dt
d
3
Consider the square matrix A whose elements are defined by the equations
dqi_ dQi
dt
> t, j = 1,2, ...,n,
and show that for the Jacobian (1.4) the following representation holds:
dt
J
dt
-det, A.
(3.5)
dq,
For this, we multiply each jth column of the determinant (1.4) by the derivative (j = = 1, 2, ..., n) and add to its (n + l)th column, and then multiply the last row of the resulting determinant by qi and subtract from the ith row (i = 1, 2, ..., n) and, taking into account the relations ^ = q^, we simplify the elements of the last column. These transformations lead to the following equations:
J
Ë11 dq. dqi
9q1 9<in dt
Êân . . Êân dQn
<9<?i d1n dt
dt dt dt
<9<?i d1n dt
d<h dq1
à M.
dg,
dq„ X dt dq, Vn dqx dt dq1
91n
Qldqn
dt
dq-j dt
ÊSn _ â _
dqn Vndqn dt d1n
d<ln A dt dt Hn dt
ik
dt
ËÈL _ à M.
dq-,
dg„ a dt dq1 Vn dq, dt dt?!
w^-QiP- 0
dq dq
- n f)
Vndqn U
dt dt
dqn dt
The expansion of the last determinant in terms of the elements of the (n + 1)th column shows the validity of the representation (3.5) and therefore condition (1.4) can be written as
dt , „ , — det A / 0. dt
Now, it immediately follows from (3.4) that the old equations (1.1) and the new equations (1.9) (with the Lagrangian function (1.10)) are equivalent.
4. Examples
1. In Eqs. (1.1) with the Lagrangian function L(qi, q_i, t) we make a change of variables (1.3) of the form
qi = at t = (it, i = 1, 2,...,n, (4.1)
where ai and ( are constants. Such a change of variables is related to the rescaling of length, time etc. and is often used to write equations of motion in dimensionless variables. From (1.5), (1.7) and (4.1) we have
dt dtt
= ß,
ai dQi • -, «
* ß dt
Equations (1.1) are transformed into Eqs. (1.9) with the new Lagrangian function L calculated by the formula
«i ^ ^ (42)
L = f3L {atqt, fr
In fact, the constant multiplier / in front of L could be dropped since two Lagrangian functions differing by a constant multiplier generate identical equations of motion.
2. Based on the remark made at the end of the preceding example, one can sometimes draw important conclusions on some properties of the motion of a material system without integrating its equations of motion.
As an example, we consider a free material point of mass m moving relative to a fixed coordinate system in a potential force field. The position of the point is specified by its Cartesian coordinates x, y, z. Let the potential energy n(x, y, z) of the point be a homogeneous function of degree k, i.e., let the following equation hold for any x, y, z:
n(ax, ay, az) = akn(x, y, z).
The Lagrangian function has the form
1
L = -m (x2 + y2 + z2) - n(.T, y, z).
(4.3)
(4.4)
We make the similarity transformation
x = ax, y = ay, z = azt, t = /t. (4.5)
From Eqs. (4.2) and (4.3) we obtain the following expression for the new Lagrangian function:
- a21 L = j2m
dx\ f dy
dt) V dt
dz f d?
- ßakn(x, y, z).
If
a
J
= ßak
(4.6)
(4.7)
then the equations of motion in the new variables z, y, z, t look exactly the same as those in the old variables x, y, z, t (see the remark at the end of Example 1).
Multiplication of the coordinates by the constant multiplier a implies a transformation of one curve into another curve geometrically similar to the former (the latter curve is homothetic to the former). And hence, if y is the trajectory of the material point, then there exists (under appropriate initial conditions) a trajectory z homothetic to it.
Let t and t be the times of motion of a material point along the arc AB of the curve y and along the arc AB of the curve t, respectively. Let d and d denote the distances of points A and A from the origin of coordinates. By virtue of (4.5) we have
t
= ß,
d
— = a. d
But from (4.7) it follows that
ß
= a(2-k)/2.
2
2
t
Therefore, the ratio of the times of motion of the material point along the corresponding segments
of the trajectories 7 and 7 is defined by the equation
t ( d \(2-k)/2
Hi) ■ (4'8)
As an illustration, we consider specific examples of using relation (4.8).
1. If there is no force field, it may be assumed that n is equal to an arbitrary constant and hence does not change under the transformations (4.5). Therefore, the quantity k should be assumed to be zero. Then it can be seen from (4.8) that the material point covers equal distances in equal time intervals. This was to be expected since in the absence of a force field the point moves uniformly in a straight line.
2. Let a point fall in a homogeneous gravitational field first from one height and then from another height. Here k = 1 and it follows from (4.8) that the ratio between the heights from which the material point falls is equal to the ratio between the squares of the corresponding times of fall.
3. For small linear oscillations k = 2 and, according to (4.8), the period of oscillations does not depend on the amplitude.
4. In the case of motion in the central Newtonian gravitational field k = —1. And equation (4.8) shows that the squares of the periods of revolution of the material point in the orbits 7 and 7 are related as the cubes of the linear dimensions of these orbits (the third Kepler law).
3. It may happen that under some specially chosen changes of variables (1.3) the La-grangian function L^qi, t^j does not change. This implies that, the new Lagrangian function is calculated by the formula
i.e., it is obtained from the old Lagrangian function by adding a tilde to all variables. In these cases Lagrange's equations are not only covariant, but also invariant under the chosen changes of variables. Such cases are of great interest in analytical dynamics and theoretical physics.
As an example, we consider a Lagrangian function of the form
The equation corresponding to this function (Emden's equation [8]) arises in the problem of the equilibrium of a polytropic gas sphere, between whose particles the forces of mutual gravitational attraction act.
Let us choose a transformation (1.3) in the form
q = aq, t = -^i, (4.9)
a2
where a is an arbitrary nonzero constant. Calculations by the formulae (1.7) and (1.10) show that the change of variables (4.9) does not change the Lagrangian function, and for any considered value of a we have
MK(f)2-H
4. Let us continue the analysis of the system from Example 2. We answer the question of whether there exists a degree of homogeneity of the potential energy n(x, y, z) such that the similarity transformation (4.5) does not change the Lagrangian function (4.4).
The answer to this question immediately follows from the expression (4.6) for the new Lagrangian function: the following equations must be satisfied:
a2 0k
— = pak = 1 /
and hence
k = -2, / = a2. (4.10)
As a concrete example, we consider the problem of the motion of a charged particle in the plane Oxy in the dipole force field. In this case the potential energy is given by the formula [9]
n ^
(x2 + y2)3/2'
where i is a constant. Here k = —2 and the change of variables
x = ax, y = ay, t = a2t, where a is a nonzero constant, does not change the Lagrangian function.
References
[1] Fichtenholz, G.M., Differential- und Integralrechnung: In 3 Vols., Hochschulbücher für Mathematik, vols. 61-63, Frankfurt am Main: Deutsch, 2006.
[2] Markeev, A. P., Theoretical Mechanics, Izhevsk: R&C Dynamics, Institute of Computer Science, 2007 (Russian).
[3] Aizerman, M. A., Classical Mechanics, Moscow: Nauka, 1980 (Russian).
[4] Yakovenko, G.N., A Short Course in Analytical Dynamics, Moscow: Binom, 2004 (Russian).
[5] Kotkin, G. L., Serbo, V. G., and Chernykh, A. I., Lectures on Analytical Mechanics, Izhevsk: R&C Dynamics, Institute of Computer Science, 2010 (Russian).
[6] Belenky, I. M., Introduction to Analytical Mechanics, Moscow: Vysshaya Shkola, 1964 (Russian).
[7] Pars, L. A., A Treatise on Analytical Mechanics, London: Heinemann, 1965.
[8] Sansone, G., Equazioni differenziali nel campo reale: Parte seconda, 2nd ed., Bologna: Zanichelli, 1949.
[9] Tamm, I. E., Fundamentals of the Theory of Electricity, Moscow: Mir, 1979.
